Complete newbie, all vertices moving instead of the 3 I selected, not proportional editing
The reason you're seeing that effect is that you have the 'On Cage' option selected for your subdivision surface modifier.



What this option effectively does it it allows you to work on the result of the subdivision surface on your mesh in edit mode, as opposed to the mesh before the subdivision surface.

Since the subdivision surface moves vertices based upon the positions of their surrounding vertices, when you move a vertex with 'On Cage' active, you also move surrounding vertices.

Simply toggle off the indicated button, and you should see the wireframe of your cube in Edit mode, and be able to move one vertex at a time again.